AN ALGORITHMIC APPROACH TO BAYESIAN LINEAR MODEL CALCULATIONS

A framework is described for organizing and understanding the computations necessary to obtain the posterior mean of a vector of linear effects in a normal linear model, conditional on the parameters that determine covariance structure. The approach has two major uses; firstly, as a pedagogical tool in the derivation of formulae, and secondly, as a practical tool for developing computational strategies without needing complicated matrix formulae that are often unwieldy in complex hierarchical models. The proposed technique is based upon symbolic application of the sweep operator SWP to an appropriate tableau of means and covariances. The method is illustrated with standard linear model specifications, including the so-called mixed model, with both fixed and random effects.     

A tutorial on adaptive MCMC

We review adaptive Markov chain Monte Carlo algorithms (MCMC) as a mean to optimise their performance. Using simple toy examples we review their theoretical underpinnings, and in particular show why adaptive MCMC algorithms might fail when some fundamental properties are not satisfied. This leads to guidelines concerning the design of correct algorithms. We then review criteria and the useful framework of stochastic approximation, which allows one to systematically optimise generally used criteria, but also analyse the properties of adaptive MCMC algorithms. We then propose a series of novel adaptive algorithms which prove to be robust and reliable in practice. These algorithms are applied to artificial and high dimensional scenarios, but also to the classic mine disaster dataset inference problem.     

A response to M. Stone: 'How not to measure the efficiency of public services (and how one might)'

Summary.  This is a response to Stone's criticisms of the Spottiswoode report to the UK Treasury which was responding to the Treasury's request for improved methods to evaluate the efficiency and productivity of the 43 police districts in England and Wales. The Spottiswoode report recommended uses of data envelopment analysis (DEA) and stochastic frontier analysis (SFA), which Stone critiqued en route to proposing an alternative approach. Here we note some of the most serious errors in his criticism and inaccurate portrayals of DEA and SFA. Most of our attention is devoted to DEA, and to Stone's recommended alternative approach without much attention to SFA, partly because of his abbreviated discussion of the latter. In our response we attempt to be constructive as well as critical by showing how Stone's proposed approach can be joined to DEA to expand his proposal beyond limitations in his formulations.     

Nonconglomerative coherent conditional probabilities in statistical inference

Noteworthy connections among conglomerability, countable additivity and coherence are discussed in detail, reaching the conclusion that nonconglomerable conditional probabilities must not be doomed and play a significant role in statistical inference. Extended and updated version of a contributed paper presented at the International Conference on “Information Processing and Management of Uncertainty in knowledge-based systems”, IPMU 2004, Perugia, Italy.     

COVARIANCE MODELS FOR LATTICE DATA

Given observations on an m × n lattice, approximate maximum likelihood estimates are derived for a family of models including direct covariance, spatial moving average, conditional autoregressive and simultaneous autoregressive models. The approach involves expressing the (approximate) covariance matrix of the observed variables in terms of a linear combination of neighbour relationship matrices, raised to a power. The structure is such that the eigenvectors of the covariance matrix are independent of the parameters of interest. This result leads to a simple Fisher scoring type algorithm for estimating the parameters. The ideas are illustrated by fitting models to some remotely sensed data.     

Analysis of growth curve data by using cubic smoothing splines

Longitudinal data frequently arises in various fields of applied sciences where individuals are measured according to some ordered variable, e.g. time. A common approach used to model such data is based on the mixed models for repeated measures. This model provides an eminently flexible approach to modeling of a wide range of mean and covariance structures. However, such models are forced into a rigidly defined class of mathematical formulas which may not be well supported by the data within the whole sequence of observations. A possible non-parametric alternative is a cubic smoothing spline, which is highly flexible and has useful smoothing properties. It can be shown that under normality assumption, the solution of the penalized log-likelihood equation is the cubic smoothing spline, and this solution can be further expressed as a solution of the linear mixed model. It is shown here how cubic smoothing splines can be easily used in the analysis of complete and balanced data. Analysis can be greatly simplified by using the unweighted estimator studied in the paper. It is shown that if the covariance structure of random errors belong to certain class of matrices, the unweighted estimator is the solution to the penalized log-likelihood function. This result is new in smoothing spline context and it is not only confined to growth curve settings. The connection to mixed models is used in developing a rough testing of group profiles. Numerical examples are presented to illustrate the techniques proposed.     

Newsvendor Problems with Demand Shocks and Unknown Demand Distributions

In today's competitive market, demand volume and even the underlying demand distribution can change quickly for a newsvendor seller. We refer to sudden changes in demand distribution as demand shocks. When a newsvendor seller has limited demand distribution information and also experiences underlying demand shocks, the majority of existing methods for newsvendor problems may not work well since they either require demand distribution information or assume stationary demand distribution. We present a new, robust, and effective machine learning algorithm for newsvendor problems with demand shocks but without any demand distribution information. The algorithm needs only an approximate estimate of the lower and upper bounds of demand range; no other knowledge such as demand mean, variance, or distribution type is necessary. We establish the theoretical bounds that determine this machine learning algorithm's performance in handling demand shocks. Computational experiments show that this algorithm outperforms the traditional approaches in a variety of situations including large and frequent shocks of the demand mean. The method can also be used as a meta‐algorithm by incorporating other traditional approaches as experts. Working together, the original algorithm and the extended meta‐algorithm can help manufacturers and retailers better adapt their production and inventory control decisions in dynamic environments where demand information is limited and demand shocks are frequent      

Managing stochastic demand in an Inventory Routing Problem with transportation procurement

We study an Inventory Routing Problem in which the supplier has a limited production capacity and the stochastic demand of the retailers is satisfied with procurement of transportation services. The aim is to minimize the total expected cost over a planning horizon, given by the sum of the inventory cost at the supplier, the inventory cost at the retailers, the penalty cost for stock-out at the retailers and the transportation cost. First, we show that a policy based just on the average demand can have a total expected cost infinitely worse than the one obtained by taking into account the overall probability distribution of the demand in the decision process. Therefore, we introduce a stochastic dynamic programming formulation of the problem that allows us to find an optimal policy in small size instances. Finally, we design and implement a matheuristic approach, integrating a rollout algorithm and an optimal solution of mixed-integer linear programming models, which is able to solve realistic size problem instances. Computational results allow us to provide managerial insights concerning the management of stochastic demand.     

Moderate Deviation Principles for Empirical Covariance in the Neighbourhood of the Unit Root

In this paper, we consider the linear autoregressive model with varying coefficients θ_n∈[0,1). When θ_n tending to the unit root, the moderate deviation principle for empirical covariance is discussed, and as statistical applications, we provide the moderate deviation estimates of the least square and the Yule–Walker estimators of the parameter θ_n.     

A useful approach to identify the multicollinearity in the presence of outliers

The presence of outliers in the data sets affects the structure of multicollinearity which arises from a high degree of correlation between explanatory variables in a linear regression analysis. This affect could be seen as an increase or decrease in the diagnostics used to determine multicollinearity. Thus, the cases of outliers reduce the reliability of diagnostics such as variance inflation factors, condition numbers and variance decomposition proportions. In this study, we propose to use a robust estimation of the correlation matrix obtained by the minimum covariance determinant method to determine the diagnostics of multicollinearity in the presence of outliers. As a result, the present paper demonstrates that the diagnostics of multicollinearity obtained by the robust estimation of the correlation matrix are more reliable in the presence of outliers.